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On the game p-Laplacian on weighted graphs with applications in image processing and data clustering

  • A. ELMOATAZ (a1), X. DESQUESNES (a2) and M. TOUTAIN (a3)

Game-theoretic p-Laplacian or normalized p-Laplacian operator is a version of classical variational p-Laplacian which was introduced recently in connection with stochastic games called Tug-of-War with noise (Peres et al. 2008, Tug-of-war with noise: A game-theoretic view of the p-laplacian. Duke Mathematical Journal 145(1), 91–120). In this paper, we propose an adaptation and generalization of this operator on weighted graphs for 1 ≤ p ≤ ∞. This adaptation leads to a partial difference operator which is a combination between 1-Laplace, infinity-Laplace and 2-Laplace operators on graphs. Then we consider the Dirichlet problem associated to this operator and we prove the uniqueness and existence of the solution. We show that the solution leads to an iterative non-local average operator on graphs. Finally, we propose to use this operator as a unified framework for interpolation problems in signal processing on graphs, such as image processing and machine learning.

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This work was funded under a doctoral grant supported by the Coeur et Cancer association, the regional council of Normandy, the European FEDER Grant (PLANUCA Project) and the project ANR GRAPHSIP.

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European Journal of Applied Mathematics
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